Practice Questions: Circular Motion and Gravitation

SECTION I: MULTIPLE CHOICE

Directions

Answer the following 20 multiple-choice questions. Each question has four answer choices labeled A through D. Select the one best answer for each question. Calculators are allowed. The acceleration due to gravity at Earth's surface is g = 9.8 m/s² unless otherwise specified.

Question 1

A small mass is attached to a string and whirled in a horizontal circle of radius 0.80 m on a frictionless table. The string exerts a tension force of 25 N on the mass, causing it to move at a constant speed of 4.0 m/s. What is the mass of the object?

1. 0.50 kg
2. 1.25 kg
3. 2.00 kg
4. 5.00 kg

Question 2

A car travels around a flat, unbanked circular curve with radius r at constant speed v. The coefficient of static friction between the tires and the road is μ_s. Which of the following expressions correctly represents the maximum speed at which the car can travel without slipping?

1. \( v_{\text{max}} = \sqrt{\mu_s g r} \)
2. \( v_{\text{max}} = \sqrt{\frac{g r}{\mu_s}} \)
3. \( v_{\text{max}} = \mu_s g r \)
4. \( v_{\text{max}} = \frac{\mu_s g}{r} \)

Question 3

Which of the following graphs best represents the relationship between centripetal acceleration a_c (vertical axis) and radius r (horizontal axis) when angular velocity is held constant?

1. A horizontal straight line
2. A straight line passing through the origin with positive slope
3. A parabola opening upward
4. A hyperbola decreasing as r increases

Question 4

A satellite orbits Earth at a distance of 2R from Earth's center, where R is Earth's radius. A second satellite orbits at a distance of 4R from Earth's center. What is the ratio of the orbital period of the second satellite to that of the first satellite?

1. √2 : 1
2. 2 : 1
3. 2√2 : 1
4. 4 : 1

Question 5

Data Table:

A student wishes to verify Kepler's Third Law using the data above. Which of the following graphical analyses would produce a straight line through the origin if the law is correct?

1. Graph of orbital period T versus orbital radius r
2. Graph of T² versus r³
3. Graph of T³ versus r²
4. Graph of ln(T) versus ln(r)

Question 6

An astronaut in a spacecraft experiences apparent weightlessness while orbiting Earth. Which of the following best explains this phenomenon?

1. The gravitational force on the astronaut is zero at orbital altitude.
2. The gravitational force is balanced by the centrifugal force pushing outward.
3. The astronaut and the spacecraft are both in free fall, accelerating toward Earth at the same rate.
4. The orbital velocity is so high that gravity cannot affect the astronaut.

Question 7

Two satellites, X and Y, orbit Earth. Satellite X has twice the mass of satellite Y and orbits at twice the radius. How does the gravitational force on satellite X compare to the gravitational force on satellite Y?

1. The force on X is half the force on Y.
2. The force on X is equal to the force on Y.
3. The force on X is twice the force on Y.
4. The force on X is four times the force on Y.

Question 8

A ball of mass m is attached to a string and swung in a vertical circle of radius r at constant speed. At which point in the circular path is the tension in the string greatest?

1. At the top of the circle
2. At the bottom of the circle
3. At the side of the circle when the ball is at the same height as the center
4. The tension is the same at all points because the speed is constant

Question 9

What is the minimum speed the cart must have at the top of the loop to maintain contact with the track?

1. \( v = \sqrt{gR} \)
2. \( v = \sqrt{2gR} \)
3. \( v = \sqrt{3gR} \)
4. \( v = \sqrt{5gR} \)

Question 10

Planet X has twice the mass of Earth and twice the radius. What is the acceleration due to gravity at the surface of Planet X compared to the acceleration due to gravity at Earth's surface?

1. \( \frac{g}{4} \)
2. \( \frac{g}{2} \)
3. g
4. 2g

Question 11

A space station is designed to create artificial gravity by rotating. The station has a radius of 100 m. At what angular velocity must the station rotate so that astronauts at the outer edge experience an apparent acceleration equal to g = 9.8 m/s²?

1. 0.098 rad/s
2. 0.31 rad/s
3. 0.98 rad/s
4. 3.1 rad/s

Question 12

Which of the following best describes the mathematical relationship shown in the graph?

1. Linear relationship: F is proportional to r
2. Inverse relationship: F is proportional to 1/r
3. Inverse square relationship: F is proportional to 1/r²
4. Exponential decay: F is proportional to e^-r

Question 13

A pendulum consists of a mass suspended by a string. When the mass swings through the lowest point of its arc at speed v, the tension in the string is T. If the speed at the lowest point is doubled to 2v, what is the new tension in the string?

1. T + mg
2. 2T
3. 3T - 2mg
4. 4T - 3mg

Question 14

Two planets have the same mass but different radii. Planet A has radius R, and Planet B has radius 2R. An object is dropped from rest at a height h above the surface of each planet (where h is much smaller than R). Which statement correctly compares the time it takes for the object to reach the surface?

1. The time is the same for both planets.
2. The time for Planet A is half the time for Planet B.
3. The time for Planet A is \( \frac{1}{\sqrt{2}} \) times the time for Planet B.
4. The time for Planet A is \( \frac{1}{2} \) times the time for Planet B.

Question 15

Which of the following correctly explains this observation?

1. As θ increases, the effective length of the pendulum decreases, reducing the period.
2. As θ increases, the centripetal acceleration increases, which increases the angular velocity and decreases the period.
3. As θ increases, the tension in the string increases, which increases the speed and decreases the period.
4. All of the above contribute to the observed relationship.

Question 16

A binary star system consists of two stars of equal mass M orbiting their common center of mass. The distance between the two stars is d. What is the period of their orbit?

1. \( T = 2\pi \sqrt{\frac{d^3}{GM}} \)
2. \( T = 2\pi \sqrt{\frac{d^3}{2GM}} \)
3. \( T = 2\pi \sqrt{\frac{d^3}{4GM}} \)
4. \( T = \pi \sqrt{\frac{d^3}{GM}} \)

Question 17

A marble rolls around the inside of a conical funnel, maintaining a constant height above the bottom. The funnel's surface makes an angle of 45° with the horizontal. If the radius of the circular path is 0.20 m, what is the speed of the marble?

1. 0.70 m/s
2. 1.4 m/s
3. 2.0 m/s
4. 2.8 m/s

Question 18

If the Earth-Moon distance is d and the mass of Earth is 81 times the mass of the Moon, what is the distance from Earth's center to point P?

1. \( \frac{d}{10} \)
2. \( \frac{9d}{10} \)
3. \( \frac{d}{2} \)
4. \( \frac{d}{9} \)

Question 19

A satellite in a circular orbit around Earth has kinetic energy K and gravitational potential energy U. Which of the following correctly relates the total mechanical energy E of the satellite?

1. E = K + U = 0
2. E = K + U = -K
3. E = K + U = -U/2
4. E = K + U = K/2

Question 20

While this model demonstrates some qualitative features of orbital motion, which of the following is a significant limitation of this analogy?

1. The model requires gravity (the vertical pull on the sheet) to work, whereas actual orbital motion occurs in the absence of a preferred direction.
2. The model shows planets spiraling inward, whereas real planets maintain stable orbits.
3. The model cannot show elliptical orbits, only circular ones.
4. The model violates conservation of angular momentum.

SECTION II: FREE RESPONSE

Directions

Answer both of the following questions. Show all your work clearly, including algebraic manipulation and numerical substitutions. In problems involving derivations, begin by writing a fundamental physics principle or defining equation. For problems involving experimental design, describe your procedure clearly enough that another student could replicate your experiment. Answers without supporting work will receive little to no credit.

FRQ 1: Mathematical Routines (Derivation and Calculation)

1. Begin your derivation by writing a fundamental physics principle or equation. Derive an expression for the maximum angular velocity ω_max at which the turntable can rotate before the block begins to slip. Express your answer in terms of r, μ_s, and fundamental constants.
2. Explain, using physics principles, why the maximum angular velocity depends on the radius r in the way shown by your derived expression.
3. A student places a block at a distance of 0.30 m from the center of the turntable. The coefficient of static friction is 0.40. Calculate the maximum angular velocity (in rad/s) at which the turntable can rotate before the block slips. Show all numerical work.
4. The student now places the same block at a distance of 0.60 m from the center. Without calculation, predict whether the maximum angular velocity will increase, decrease, or remain the same. Justify your answer using the physics of circular motion.

FRQ 2: Experimental Design and Analysis

1. Describe a procedure the students could use to test Kepler's Third Law using the available data. Your description should include:
   • What quantities should be calculated or manipulated from the raw data
   • How to organize the data for analysis
   • What measurements or comparisons will be made
2. Describe how the students should graph their data to produce a linear relationship if Kepler's Third Law is correct. Specifically:
   • Identify what quantity should be plotted on the vertical axis
   • Identify what quantity should be plotted on the horizontal axis
   • Explain what the slope of the best-fit line represents physically
3. The students collect the following data:
   
   Complete the table by calculating T² and r³ for each planet. Show a sample calculation for one planet.
4. On the grid below, plot the data points and draw a best-fit line through the data. Label your axes with quantities and units.

   [Grid for student plotting]

5. Use your graph to determine the value of the constant k in Kepler's Third Law. Show your calculation, including how you determined the slope and what it represents.
6. A newly discovered exoplanet orbits a distant star with a period of 2.5 Earth years. The star has the same mass as our Sun. Use Kepler's Third Law to calculate the orbital radius of this exoplanet in AU. Show all work.

ANSWER KEY

Part A - Multiple Choice Answer Table

Part B - Free Response Detailed Answers

FRQ 1 - Answer Key

Part A: Derivation of maximum angular velocity

Step 1: Begin with Newton's Second Law for circular motion.
The net force toward the center provides the centripetal force:
\[ F_{\text{net}} = ma_c \]

Step 2: Identify the forces acting on the block.
The only horizontal force is static friction, which must provide the centripetal force:
\[ f_s = ma_c \]

Step 3: Write the expression for maximum static friction.
\[ f_{s,\text{max}} = \mu_s N = \mu_s mg \]
where N = mg because the turntable is horizontal and there is no vertical acceleration.

Step 4: Write the expression for centripetal acceleration.
For circular motion with angular velocity ω:
\[ a_c = \omega^2 r \]

Step 5: Set maximum friction equal to the required centripetal force.
\[ \mu_s mg = m \omega_{\text{max}}^2 r \]

Step 6: Solve for ω_max.
\[ \omega_{\text{max}}^2 = \frac{\mu_s g}{r} \]
\[ \omega_{\text{max}} = \sqrt{\frac{\mu_s g}{r}} \]

Final Answer: \( \boxed{\omega_{\text{max}} = \sqrt{\frac{\mu_s g}{r}}} \)

Part B: Explanation of radius dependence

The maximum angular velocity decreases as the radius r increases because the centripetal acceleration required to keep the block moving in a circle is proportional to ω²r. For a given angular velocity, a larger radius requires a larger centripetal force. Since the maximum static friction force is constant (it depends only on μ_s and mg, not on r), the angular velocity must decrease as r increases to keep the required centripetal force within the available friction limit. Mathematically, ω_max is proportional to 1/√r, showing an inverse square root relationship with radius.

Part C: Numerical calculation

Given:
r = 0.30 m
μ_s = 0.40
g = 9.8 m/s²

Calculate ω_max:
\[ \omega_{\text{max}} = \sqrt{\frac{\mu_s g}{r}} \]
\[ \omega_{\text{max}} = \sqrt{\frac{(0.40)(9.8 \text{ m/s}^2)}{0.30 \text{ m}}} \]
\[ \omega_{\text{max}} = \sqrt{\frac{3.92 \text{ m/s}^2}{0.30 \text{ m}}} \]
\[ \omega_{\text{max}} = \sqrt{13.07 \text{ s}^{-2}} \]
\[ \omega_{\text{max}} = 3.6 \text{ rad/s} \]

Final Answer: 3.6 rad/s

Part D: Prediction for doubled radius

Prediction: The maximum angular velocity will decrease.

Justification: According to the derived expression \( \omega_{\text{max}} = \sqrt{\frac{\mu_s g}{r}} \), the maximum angular velocity is inversely proportional to the square root of the radius. When the radius doubles from 0.30 m to 0.60 m, the maximum angular velocity decreases by a factor of √2, becoming approximately 2.5 rad/s. This occurs because at a larger radius, the same angular velocity produces a greater centripetal acceleration, requiring more force to maintain circular motion. Since the friction force is limited, the angular velocity must decrease.

FRQ 2 - Answer Key

Part A: Experimental procedure

A complete procedure should include the following elements:

• Data collection: Gather published data for each planet showing its orbital period T (in Earth years) and average orbital radius r (in AU).
• Data manipulation: For each planet, calculate T² (square the period) and r³ (cube the radius). Record these calculated values in a data table alongside the original measurements.
• Analysis method: According to Kepler's Third Law, T² should be proportional to r³. To test this, either:
  • Calculate the ratio T²/r³ for each planet and verify that it is approximately constant, or
  • Create a graph plotting T² versus r³ and verify that the relationship is linear with a y-intercept of zero
• Verification: If the ratio is constant within experimental uncertainty, or if the graph is linear through the origin, Kepler's Third Law is supported by the data.

Part B: Graphing procedure and slope interpretation

Vertical axis: T² (period squared), measured in years²

Horizontal axis: r³ (radius cubed), measured in AU³

Slope interpretation: The slope of the best-fit line represents the constant k in Kepler's Third Law. Mathematically, since \( T^2 = kr^3 \), a graph of T² versus r³ has slope = k. Physically, this constant depends on the mass of the central body (the Sun). For our solar system, k ≈ 1.0 years²/AU³ when using these units. The slope has units of years²/AU³.

Part C: Data table completion

Sample calculation for Earth:
T² = (1.00 years)² = 1.00 years²
r³ = (1.00 AU)³ = 1.00 AU³

Complete table:

Part D: Graph construction

Students should plot the five data points from the completed table with r³ on the horizontal axis and T² on the vertical axis. The points should lie very close to a straight line passing through the origin. The best-fit line should be drawn to pass as close as possible to all data points while going through (0,0). Axes must be labeled: horizontal axis as "r³ (AU³)" and vertical axis as "T² (years²)".

Part E: Determining the constant k

Method: The constant k equals the slope of the best-fit line on the T² versus r³ graph.

Calculation using two points:
Select the point for Jupiter: (r³ = 141 AU³, T² = 141 years²)
Select the point for Earth: (r³ = 1.00 AU³, T² = 1.00 years²)

\[ k = \text{slope} = \frac{\Delta T^2}{\Delta r^3} = \frac{141 \text{ years}^2 - 1.00 \text{ years}^2}{141 \text{ AU}^3 - 1.00 \text{ AU}^3} \]
\[ k = \frac{140 \text{ years}^2}{140 \text{ AU}^3} = 1.0 \text{ years}^2/\text{AU}^3 \]

Physical meaning: The constant k = 1.0 years²/AU³ for our solar system. This value is specific to the Sun's mass and the units chosen.

Final Answer: k = 1.0 years²/AU³

Part F: Application to exoplanet

Given:
T = 2.5 Earth years
Star mass = Sun's mass, so k = 1.0 years²/AU³
Find: orbital radius r in AU

Use Kepler's Third Law:
\[ T^2 = kr^3 \]

Solve for r:
\[ r^3 = \frac{T^2}{k} \]
\[ r = \left(\frac{T^2}{k}\right)^{1/3} \]

Substitute values:
\[ r = \left(\frac{(2.5 \text{ years})^2}{1.0 \text{ years}^2/\text{AU}^3}\right)^{1/3} \]
\[ r = \left(\frac{6.25 \text{ years}^2}{1.0 \text{ years}^2/\text{AU}^3}\right)^{1/3} \]
\[ r = (6.25 \text{ AU}^3)^{1/3} \]
\[ r = 1.8 \text{ AU} \]

Final Answer: The exoplanet orbits at a distance of 1.8 AU from its star.